Automorphisms of partition order-decreasing transformation monoids.
Let \(T_n\) (\(S_n\)) be the full transformation semigroup (the symmetric group, respectively) on an \(n\)-element set \(X_n\), \(\rho\) an equivalence relation on \(X_n\), \(\preceq\) a total order on \(X_n/\rho\), \(T(\rho,\preceq)=\{\alpha\in T_n:(x\alpha)\rho\preceq x\rho,\;\forall x\in X_n\}\) and \(U_\rho=\{\mu\in S_n:(x\rho)\mu=x\rho,\;x\in X_n\}\). The main theorem states that \(U_\rho\) is the group of units of \(T(\rho,\preceq)\) and for any \(\varphi\in\Aut\,T(\rho,\preceq)\), there exists \(\mu\in U_\rho\) such that \((\alpha)\varphi=\mu^{-1}\alpha\mu\) for all \(\alpha\in T(\rho,\preceq)\). Conversely, if \(\mu\in U_\rho\), then \(\mu^{-1}\alpha\mu\in T(\rho,\preceq)\) for any \(\alpha\in T(\rho,\preceq)\), and the map \(\varphi\colon T(\rho,\preceq)\to T(\rho,\preceq)\) defined by \((\alpha)\varphi=\mu^{-1}\alpha\mu\), \(\alpha\in T(\rho,\preceq)\), is an automorphism.
partition order-decreasingg transformations
scientific article; zbMATH DE number 6140995
scientific article